This theorem implies that the only way a language can be incomplete is [that] there is a model of the language in which a particular statement is true, and another in which the statement is false.

For example, we can see that [for] the language [comprising] the symbols 0, 1, +, -, and , [] the statement []   is true [] if we take the structure to be C or R, but not if we choose Q. So it is clear that the formula a*a = 2 is not true in every model of the language and the thus the language is incomplete.

What the completeness theorem asserts is that this is the only way that a theory (set of formulas) can be incomplete and that every formula that satisfies every structure is provable in the language.

— Godel’s Completeness And Incompleteness Theorems

— Ben Chaiken

2012.12.04 Tuesday ACHK